Internal Conflict-Free Projection Sets

  • Łukasz Mikulski
Conference paper
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 59)


Projections into free monoids are one of possible ways of describing finite traces. That approach is used to expand them to infinite traces, which are defined as members of a proper subfamily of cartesian product of mentioned free monoids, completed with least upper bounds in the Scott’s sense. Elements of the cartesian product are called projection sets and elements of the proper subfamily are called reconstructible projection sets. The proposed description gives useful methods of investigation. Especially, it contains a constructive operation that turns an arbitrary element of cartesian product to the closest trace. Then, we define constructively a proper subclass of projection sets, called internal conflict-free projection sets, and show using structural induction that it is precisely equal to subfamily of reconstructible projection sets. This is the main result of the paper. The proofs are ommited becouse of the page limit.


projection sets free monoids structural induction finite traces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cartier, P., Foata, D.: Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics, vol. 85. Springer, Berlin (1969)zbMATHGoogle Scholar
  2. 2.
    Diekert, V.: Combinatorics on Traces. Springer, Berlin (1990)zbMATHGoogle Scholar
  3. 3.
    Diekert, V., Rozenberg, G. (eds.): The Book of Traces. World Scientific, Singapore (1995)Google Scholar
  4. 4.
    Gastin, P.: Infinite traces. In: Guessarian, I. (ed.) LITP 1990. LNCS, vol. 469, pp. 277–308. Springer, Heidelberg (1990)Google Scholar
  5. 5.
    Gastin, P., Petit, A.: Infinite traces. In: Diekert, V., Rozenberg, G. (eds.) The Book of Traces, ch. 11, pp. 393–486. World Scientific, Singapore (1995)Google Scholar
  6. 6.
    Kwiatkowska, M.Z.: Defining process fairness for non-interleaving concurrency. Foundations of Software Technology and Theoretical Computer Science 10, 286–300 (1990)MathSciNetGoogle Scholar
  7. 7.
    Mazurkiewicz, A.: Concurrent program schemes and their interpretations. Daimi report pb-78, Aarhus University (1977)Google Scholar
  8. 8.
    Mikulski, Ł.: Projection representation of Mazurkiewicz traces. Fundamenta Informaticae 85, 399–408 (2008)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Shields, M.W.: Concurrent machines. The Computer Journal 28(5), 449–465 (1985)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Łukasz Mikulski
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityTorun̈Poland

Personalised recommendations